Given: Perpendicular dropped from A(7, 14, 5) on to the plane 2x + 4y – z = 2

To find: co-ordinates of the foot of perpendicular

Formula Used: Equation of a line is

Where b₁:b₂:b₃ is the direction ratio and (x₁, x₂, x₃) is a point on the line.

Explanation:

Let the foot of the perpendicular be (a, b, c)

Since this point lies on the plane,

2a + 4b – c = 2 … (1)

Direction ratio of the normal to the plane is 2 : 4 : -1

Direction ratio perpendicular = direction ratio of normal to the plane

So, equation of the perpendicular is

Since (a, b, c) is a point on the perpendicular,

(7, 14, 5) is a point on the perpendicular.

So, a = 7 - 2λ, b = 14 – 4λ, c = 5 + λ

Substituting in (1),

14 – 4λ + 56 – 16λ – 5 - λ = 2

21λ = 70 – 7 = 63

λ = 3

Therefore, foot of the perpendicular is (1, 2, 8)